metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊1F5, C40⋊1C4, D5.1D8, D10.9D4, D5.1Q16, Dic5.3Q8, C5⋊(C2.D8), C5⋊2C8⋊6C4, C4⋊F5.4C2, C4.9(C2×F5), C20.9(C2×C4), (C8×D5).3C2, C2.5(C4⋊F5), C10.2(C4⋊C4), (C4×D5).26C22, SmallGroup(160,69)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5.D8
G = < a,b,c,d | a5=b2=c8=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a2b, dcd-1=c-1 >
Character table of D5.D8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 10 | 20A | 20B | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 5 | 5 | 2 | 10 | 20 | 20 | 20 | 20 | 4 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | -1 | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | -i | i | i | -i | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | i | -i | -i | i | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | √2 | -√2 | -2 | 0 | 0 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | -√2 | √2 | -2 | 0 | 0 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | -√2 | √2 | -2 | 0 | 0 | √2 | √2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | √2 | -√2 | -2 | 0 | 0 | -√2 | -√2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ15 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ16 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ17 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√-5 | √-5 | √-5 | -√-5 | complex lifted from C4⋊F5 |
| ρ18 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √-5 | -√-5 | -√-5 | √-5 | complex lifted from C4⋊F5 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -2√2 | 2√2 | 0 | 0 | 1 | √-5 | -√-5 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2√2 | -2√2 | 0 | 0 | 1 | -√-5 | √-5 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2√2 | -2√2 | 0 | 0 | 1 | √-5 | -√-5 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -2√2 | 2√2 | 0 | 0 | 1 | -√-5 | √-5 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | complex faithful |
►On 40 points
Generators in S40
(1 26 37 18 10)(2 27 38 19 11)(3 28 39 20 12)(4 29 40 21 13)(5 30 33 22 14)(6 31 34 23 15)(7 32 35 24 16)(8 25 36 17 9)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)
(1 8)(2 7)(3 6)(4 5)(9 18 25 37)(10 17 26 36)(11 24 27 35)(12 23 28 34)(13 22 29 33)(14 21 30 40)(15 20 31 39)(16 19 32 38)
G:=sub<Sym(40)| (1,26,37,18,10)(2,27,38,19,11)(3,28,39,20,12)(4,29,40,21,13)(5,30,33,22,14)(6,31,34,23,15)(7,32,35,24,16)(8,25,36,17,9), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40), (1,8)(2,7)(3,6)(4,5)(9,18,25,37)(10,17,26,36)(11,24,27,35)(12,23,28,34)(13,22,29,33)(14,21,30,40)(15,20,31,39)(16,19,32,38)>;
G:=Group( (1,26,37,18,10)(2,27,38,19,11)(3,28,39,20,12)(4,29,40,21,13)(5,30,33,22,14)(6,31,34,23,15)(7,32,35,24,16)(8,25,36,17,9), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40), (1,8)(2,7)(3,6)(4,5)(9,18,25,37)(10,17,26,36)(11,24,27,35)(12,23,28,34)(13,22,29,33)(14,21,30,40)(15,20,31,39)(16,19,32,38) );
G=PermutationGroup([(1,26,37,18,10),(2,27,38,19,11),(3,28,39,20,12),(4,29,40,21,13),(5,30,33,22,14),(6,31,34,23,15),(7,32,35,24,16),(8,25,36,17,9)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40)], [(1,8),(2,7),(3,6),(4,5),(9,18,25,37),(10,17,26,36),(11,24,27,35),(12,23,28,34),(13,22,29,33),(14,21,30,40),(15,20,31,39),(16,19,32,38)])
D5.D8 is a maximal subgroup of
C80⋊2C4 C80⋊3C4 D5.D16 D5.Q32 (C2×C8)⋊6F5 M4(2)⋊1F5 D8×F5 SD16⋊F5 Q16×F5 Dic5.4Dic6 D5.D24
D5.D8 is a maximal quotient of
C80⋊2C4 C80⋊3C4 C16.F5 C80.2C4 C40⋊1C8 Dic5.13D8 D10.10D8 Dic5.4Dic6 D5.D24
Matrix representation of D5.D8 ►in GL4(𝔽7) generated by
| 3 | 1 | 3 | 3 |
| 0 | 1 | 6 | 3 |
| 4 | 4 | 2 | 2 |
| 1 | 5 | 6 | 0 |
| 6 | 0 | 4 | 2 |
| 0 | 6 | 1 | 4 |
| 0 | 3 | 3 | 1 |
| 0 | 1 | 3 | 6 |
| 3 | 3 | 0 | 2 |
| 1 | 3 | 1 | 1 |
| 2 | 2 | 6 | 3 |
| 3 | 4 | 2 | 3 |
| 4 | 1 | 3 | 3 |
| 6 | 2 | 6 | 6 |
| 1 | 2 | 1 | 2 |
| 0 | 2 | 1 | 0 |
G:=sub<GL(4,GF(7))| [3,0,4,1,1,1,4,5,3,6,2,6,3,3,2,0],[6,0,0,0,0,6,3,1,4,1,3,3,2,4,1,6],[3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[4,6,1,0,1,2,2,2,3,6,1,1,3,6,2,0] >;
D5.D8 in GAP, Magma, Sage, TeX
D_5.D_8
% in TeX
G:=Group("D5.D8"); // GroupNames label
G:=SmallGroup(160,69);
// by ID
G=gap.SmallGroup(160,69);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,151,579,69,2309,1169]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^8=1,d^2=a^-1*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of D5.D8 in TeX
Character table of D5.D8 in TeX